Our first problem of study in this paper, in the tradition of the seminal paper of [1]}, is
to find conditions under which stable outcomes exist for bilateral sharing, and can be found in polynomial time. We model bilateral sharing as a general network/graph formation game, similar to [2]} and [3]}. This game is a more general version of hedonic games that are known not to have core stable outcomes in general [4]}. We also consider the notion of strong stability. In this case, without imposing a utility structure on the agents, we find conditions under which the game has a strongly stable outcome and we design an algorithm that can find such outcomes in \(O(N^2)\)  time, where \(N\)  is the number of agents. Our framework weakens some of the unrealistic assumptions that have traditionally been imposed in the literature on strongly stable outcomes [5]}; in particular, we allow cyclicity in the trades and limited complementarity. We achieve this weakening by isolating a property—the top agent property—which defines an ordering for agents that is based on how much each agent is preferred by all other agents. Limited complementarity then arises by ensuring that no subset of agents ranked below an agent can be preferred to that agent. Our notion of top agent property has similarities to that of ordered coalitions in [4]}, the notion of the common ranking property [7]}, [8]} and the top coalition property [9]} used in establishing the existence of core stable outcomes in hedonic games.
